Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STABILITY OF A FUNCTIONAL EQUATION DERIVING FROM QUARTIC AND ADDITIVE FUNCTIONS

  • Received : 2008.10.12
  • Published : 2010.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation f(2x + y) + f(2x - y) = 4(f(x + y) + f(x - y)) - $\frac{3}{7}$(f(2y) - 2f(y)) + 2f(2x) - 8f(x).

Keywords

References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
  2. L. Cadariu, Fixed points in generalized metric space and the stability of a quartic functional equation, Bul. Stiint. Univ. Politeh. Timis. Ser. Mat. Fiz. 50(64) (2005), no. 2, 25-34.
  3. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86. https://doi.org/10.1007/BF02192660
  4. J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bull. Korean Math. Soc. 40 (2003), no. 4, 565-576. https://doi.org/10.4134/BKMS.2003.40.4.565
  5. M. Eshaghi-Gordji, A. Ebadian, and S. Zolfaghari, Stability of a functional equation deriving from cubic and quartic functions, Abstract and Applied Analysis 2008 (2008), Article ID 801904, 17 pages.
  6. M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, M. S. Moslehian, and S. Zolfaghari, Stability of a mixed type additive, quadratic, cubic and quartic functional equation, To appear.
  7. M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park, and S. Zolfaghari, Stability of an additive-cubic-quartic functional equation, Submitted.
  8. M. Eshaghi-Gordji, C. Park, and M. Bavand-Savadkouhi, Stability of a quartic type functional equation, Submitted.
  9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
  10. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  11. A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen 48 (1996), no. 3-4, 217-235.
  12. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser Boston, Inc., Boston, MA, 1998.
  13. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  14. G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of ${\psi}$-additive mappings, J. Approx. Theory 72 (1993), no. 2, 131-137. https://doi.org/10.1006/jath.1993.1010
  15. G. Isac and Th. M. Rassias, Stability of ${\psi}$-additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228. https://doi.org/10.1155/S0161171296000324
  16. S. H. Lee, S. M. Im, and I. S. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), no. 2, 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
  17. A. Najati, On the stability of a quartic functional equation, J. Math. Anal. Appl. 340 (2008), no. 1, 569-574. https://doi.org/10.1016/j.jmaa.2007.08.048
  18. C. G. Park, On the stability of the orthogonally quartic functional equation, Bull. Iranian Math. Soc. 31 (2005), no. 1, 63-70.
  19. W. G. Park and J. H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal. 62 (2005), no. 4, 643-654. https://doi.org/10.1016/j.na.2005.03.075
  20. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, J. Indian Math. Soc. (N.S.) 67 (2000), no. 1-4, 169-178.
  21. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) (1999), no. 2, 243-252.
  22. Th. M. Rassias, Functional Equations and Inequalities, Mathematics and its Applications, 518. Kluwer Academic Publishers, Dordrecht, 2000.
  23. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130. https://doi.org/10.1023/A:1006499223572
  24. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284. https://doi.org/10.1006/jmaa.2000.7046
  25. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.2307/2042795
  26. K. Ravi and M. Arunkumar, Hyers-Ulam-Rassias stability of a quartic functional equation, Int. J. Pure Appl. Math. 34 (2007), no. 2, 247-260.
  27. E. Thandapani, K. Ravi, and M. Arunkumar, On the solution of the generalized quartic functional equation, Far East J. Appl. Math. 24 (2006), no. 3, 297-312.
  28. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York 1964.

Cited by

  1. General Cubic-Quartic Functional Equation vol.2011, 2011, https://doi.org/10.1155/2011/463164
  2. FIXED POINTS AND APPROXIMATELY C*-TERNARY QUADRATIC HIGHER DERIVATIONS vol.10, pp.10, 2013, https://doi.org/10.1142/S021988781320017X
  3. Fuzzy Stability of Generalized Mixed Type Cubic, Quadratic, and Additive Functional Equation vol.2011, pp.1, 2011, https://doi.org/10.1186/1029-242X-2011-95
  4. Nearly Radical Quadratic Functional Equations inp-2-Normed Spaces vol.2012, 2012, https://doi.org/10.1155/2012/896032
  5. APPROXIMATELY QUINTIC AND SEXTIC MAPPINGS ON THE PROBABILISTIC NORMED SPACES vol.49, pp.2, 2012, https://doi.org/10.4134/BKMS.2012.49.2.339
  6. Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces vol.2012, 2012, https://doi.org/10.1155/2012/953938
  7. Approximate mixed type additive and quartic functional equation vol.35, pp.1, 2017, https://doi.org/10.5269/bspm.v35i1.29014
  8. A Fixed Point Approach to Stability of Quintic Functional Equations in Modular Spaces vol.55, pp.2, 2015, https://doi.org/10.5666/KMJ.2015.55.2.313
  9. Solution and stability of generalized mixed type additive and quadratic functional equation in non-Archimedean spaces vol.58, pp.1, 2012, https://doi.org/10.1007/s11565-011-0138-0
  10. On the stability of a mixed type quadratic-additive functional equation vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1847-2013-198
  11. Approximately Quintic and Sextic Mappings Formr-Divisible Groups into Ŝerstnev Probabilistic Banach Spaces: Fixed Point Method vol.2011, 2011, https://doi.org/10.1155/2011/572062
  12. Non-Archimedean Hyers-Ulam-Rassias stability of m-variable functional equation vol.2012, pp.1, 2012, https://doi.org/10.1186/1687-1847-2012-111
  13. A GENERALIZED ADDITIVE-QUARTIC FUNCTIONAL EQUATION AND ITS STABILITY vol.52, pp.6, 2015, https://doi.org/10.4134/BKMS.2015.52.6.1759
  14. A Functional equation related to inner product spaces in non-archimedean normed spaces vol.2011, pp.1, 2011, https://doi.org/10.1186/1687-1847-2011-37
  15. Orthogonality and quintic functional equations vol.29, pp.7, 2013, https://doi.org/10.1007/s10114-013-1061-3
  16. A FIXED-POINT METHOD FOR PERTURBATION OF HIGHER RING DERIVATIONS IN NON-ARCHIMEDEAN BANACH ALGEBRAS vol.08, pp.07, 2011, https://doi.org/10.1142/S021988781100583X
  17. A functional equation related to inner product spaces in non-Archimedean L-random normed spaces vol.2012, pp.1, 2012, https://doi.org/10.1186/1029-242X-2012-168
  18. Nearly Partial Derivations on Banach Ternary Algebras vol.6, pp.4, 2010, https://doi.org/10.3844/jmssp.2010.454.461
  19. Fixed points and approximately heptic mappings in non-Archimedean normed spaces vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1847-2013-209
  20. Stability of a quartic functional equation vol.20, pp.4, 2018, https://doi.org/10.1007/s11784-018-0629-z